You can find all the solutions by using Solve.
Solve[root^2 == 3 - 4 I, root]
{{root -> -2 + I}, {root -> 2 - I}}
If you ask for the value of a function that is multivalued then Mathematica
returns the "principal value" and this is determined by picking (arbitrary)
branch cuts. This is discussed in:
tutorial/FunctionsThatDoNotHaveUniqueValues
However, in a sense, branch lines are an artifact of insisting on a single
principal value. Riemann showed that one can construct a surface, now called
a Riemann surface, on which a multiple valued complex function is single
valued and continuous, and on which all the multiple values are recovered by
moving around the surface. The critical points (0 for the square root
function) are real but the branch cuts disappear.
One can simulate this along a parameterized line by calculating all the
roots, remembering the most recent values, and maintaining continuity of the
roots along a path by reordering if necessary.
The Presentations application has facility for this in the Multivalues
section of the ComplexGraphics sub-package. The Sqrt example is included in
the package (under the "Writing Presentations" essay). There the Sqrt
function is represented by a locator that can be dragged around the
projection of the Riemann surface onto the complex plane. The value of the
complex function is represented by an arrow attached to the point. If one
drags the point around the origin the value of the function is perfectly
smooth - the arrow never jumps discontinuously. However, it is necessary to
drag the locator around the origin twice to return to the initial value,
which indicates that we are moving on an (invisible) Riemann surface.
Murray Eisenberg and I wrote a paper, "Visualizing Complex Functions with
the Presentations Application" in The Mathematica Journal, which contained a
Riemann surface plot for:
Sqrt[z - 1] Power[z - I, (3)^-1]
This contains critical points at 1 and I. If we drag the locator around the
plane the arrow, representing the function value, varies perfectly smoothly.
If we circle, close in, the critical point at 1 we need two circuits to
return to the original value. Circling the critical point at I requires 3
circuits. Circling both critical points requires 6 circuits. However, the
Riemann surface about 1 actually has three leafs. If we repeatedly circle 1
we remain on one leaf, but if we detour to circle the critical point at I
and then return to circling 1 we will be on a different leaf and this will
be represented in the arrow values. Similarly there are two leafs to the
surface about the critical point at I and we can switch between them by
detouring around the critical point at 1.
Some functions, such as log, have an infinite number of multivalues and in
such cases we would only be able to explore a portion of the surface by this
technique. Nevertheless, Riemann surfaces are an actual practical method for
exploring and visualizing complex functions in Mathematica.
David Park
djmpark at comcast.net
http://home.comcast.net/~djmpark/index.html
From: Jacare Omoplata [mailto:walkeystalkey at gmail.com]
Hi,
When I try to find the square root of of a complex number, I get only one
answer.
In[1]:= Sqrt[3-4 I]
Out[1]= 2-I
But -2+I is an answer as well.
In[2]:= (-2+I)^2
Out[2]= 3-4 I
Why does Mathematica give the first answer and not the second? Does it
choose the answer with the positive real number? Is there any way I can get
both answers? Or do I just have to remember that the negative of the given
answer is also an answer?
Thanks.